Gravitational potential is the potential energy associated with a body that has mass. The gravitational potential obeys the principle of superposition: the gravitational potential of a collection of masses is the sum of the gravitational potentials of the individual masses. Therefore a measurement of the gravitational potential field is fundamentally a measure of the composite (additive) contributions from the mass distribution within a detectable range. The gravitational potential at a point near a gravitating body can be measured by observing the acceleration that the potential field causes on a test body, if the test body is unconstrained, i.e., if the test body is allowed to accelerate freely within the potential field. Alternatively the test body can be prevented from accelerating freely by applying a restraining force on the test body, for example through a physical spring. Using Newton's second law, the force required to restrain the test body and the mass of the test body can be used to determine the acceleration that would be observed if the test body were free to accelerate. In geophysics the term gravimeter is used for the class of instruments used to measure the gravitational acceleration to the high resolution and accuracy required for geophysical work. Since the gravitational potential field, and hence the gravitational acceleration field, depends only on the mass distribution, it is in principle possible to determine characteristics of the mass distribution through a series of measurements taken at numerous locations. However, there are several shortcomings in the use of such surveys, as described in greater detail below.
In addition to measuring the gravitational acceleration, it is also possible to measure directly at an observation point, the spatial variation in the gravitational acceleration, i.e. to measure the rate of change in the gravitational acceleration with position. The rate of change in the gravitational acceleration with change in spatial location is termed the gravity gradient. As is the case for the gravitational acceleration, in principle the mass distribution can be determined from a survey of the gravity gradient. As will be shown, there are practical advantages to using the gravity gradient for determining the mass distribution compared to using the gravitational acceleration. Gravity gradiometers have been and are used in geological surveys to measure gravity gradients as an aid to locating anomalies in the gravitational field that may be indicative of commercially valuable resources, such as minerals, precious metals and gems and oil and gas fields.
As is commonly known, the gravitational potential field at a point depends on the summed effect of all gravitating bodies and is generally characterized by the acceleration that the gravitational potential field causes on a test body. The test body is typically sized such that its mass is very small, in the limit zero, compared to the mass of the bodies causing the acceleration. The gravitational acceleration can be measured using any one of a number of existing sensors. The relationship between the gravitational acceleration and the mass of the body, or bodies, causing the acceleration was first defined by Sir Isaac Newton more than three hundred years ago. As determined by Newton, the gravitational acceleration due to a body is given by:                               a          ⁡                      (            r            )                          =                              G            *            M                                r            2                                              (        1        )            
where a is the acceleration at distance, r, from the center of mass of the gravitating body, M is the mass of the gravitating body, and G is the universal gravitational constant. This equation shows that the observed gravitational acceleration decreases with the square of the distance from the center of mass of the gravitating body. This equation holds for the acceleration due to a large, perfectly spherical body, in which case the acceleration will be directed towards the center of mass of the body which for a spherical body of uniform density will coincide with the geometric center of the body. At a point outside the surface of a large body such as the Earth, M would be the total mass of the-body, and r is the distance from the center of mass of the body. If r is smaller than the radius of the spherical body, then the mass to be considered will be the mass inside a sphere of radius r.
For a body that is not of uniform density, the gravitational acceleration is given by:                               a          ⁡                      (                          x              ,              y              ,              z                        )                          :=                  ∫                      ∫                          ∫                                                G                  ·                                                                                    ρ                        ⁡                                                  (                                                      X                            ,                            Y                            ,                            Z                                                    )                                                                    ·                      r                                                                                      (                                                                            r                                                                          )                                            3                                                                      ⁢                                                                   ⁢                                  ⅆ                  X                                ⁢                                  ⅆ                  Y                                ⁢                                  ⅆ                  Z                                                                                        (2a)             r:=(x−X)·i+(y−Y)·j+(z−Z)·k  (2b)
where the a(x,y,z) is the acceleration vector, determined at location (x,y,z) at the point of observation or measurement, ρ(X,Y,Z) is the mass density, which, in general depends on the position (X, Y, Z) within the gravitating body, r is the vector between observation point (x,y,z) and position (X, Y, Z) within the gravitating body, and i, j, and k are the unit vectors. The integrals are evaluated to include the whole volume of the gravitating body. While the contribution to the acceleration made by each infinitesimally small volume will be directed towards that volume, that is along the vector r from the observation point to the center of mass for that volume, the total acceleration at a point (x,y,z) will, in general, not be directed towards the center of mass of the entire body which, in turn, will not necessarily coincide with the geometric center as defined by the surface bounding the whole body. This is due to the r squared dependence of the acceleration, which gives greater weight to mass elements that are closer to the point at which the acceleration is determined than for mass elements that are further away.
For large gravitating bodies such as the Earth, the center of mass will be close to the geometric center. However, even when the center of mass of a body such as the Earth is coincident with its geometric center, the slight bulge of the Earth at the equator compared to the poles produces an acceleration that is not directed towards the center of mass of the Earth, except at points that are either directly over the poles or directly over the equator. This, again, arises from the r squared relationship. Hence if the gravitational acceleration is determined with respect to a coordinate system defined with respect to the geometric shape of the Earth, the gravitational acceleration will have three components, a large component directed towards the geometric center, because the geometric center is close the center of mass, and small components in the two other orthogonal directions.
As indicated by equations (2a) and (2b) above, the gravitational acceleration depends on the distribution of mass throughout the body. Therefore, in principle, it is possible to determine the density distribution of a large body by measuring the acceleration at a great number of points outside the surface of the body. For example, the acceleration of satellites orbiting the Earth can be observed using ground based optical or radio telescopes. This approach has been used to map the gravitational acceleration field of the Earth and to identify anomalies related to mass anomalies, and similarly, the motions of satellites around the Moon have been used to determine the acceleration field of the Moon and to identify mass anomalies within the Moon. The satellite observations have been used to determine variations in the Earth's acceleration field and to refine the geode models over spatial scales from approximately 100 km and greater. Since satellites typically orbit at altitudes that are several hundred km above the Earth's surface, it is not practically possible to determine variations at smaller scales.
Because of the r squared dependence, near the surface of the Earth, variations in the gravitational acceleration can be detected and related to the detailed distribution of density near the surface. The gravitational acceleration due to the mass contained within a small volume of the Earth near the Earth's surface, which is given by the density times the volume, decreases as the square of the distance from that volume element. While the gravitational acceleration is always dependent on the integral over the complete volume, variation in the acceleration field near the surface will be related primarily to density variation just below the surface. Therefore, these variations can, in principle, be used to infer the density distribution just below the surface.
The equations previously referred to provide a direct means for determining the acceleration field, that is a(x,y,z), given the description of the three dimensional density field, ρ(X,Y,Z). This is termed the forward calculation, and, if the distribution of the density is known, is straightforward. Inverting this calculation, that is determining the three dimensional density distribution from a measured acceleration field, requires solving equation (2a) above for the density given the variation of the acceleration a(x,y,z). While this is in principle possible, equation (2a) is what is known as an integral equation, and it is generally recognized as being a much more challenging problem. For practical applications, equation (2a) above can be approximated using a finite difference approach as follows:                               a          ⁡                      (                          x              ,              y              ,              z                        )                          :=                              ∑            i                                                           ⁢                                    ∑              j                                                                     ⁢                                          ∑                k                                                                               ⁢                                                G                  ·                                                                                    ρ                        ⁡                                                  (                                                                                    X                              i                                                        ,                                                          Y                              j                                                        ,                                                          Z                              k                                                                                )                                                                    ·                      r                                                                                      (                                                                            r                                                                          )                                            3                                                        ·                  Δ                                ⁢                                                                   ⁢                                                      X                    i                                    ·                  Δ                                ⁢                                                                   ⁢                                                      Y                    j                                    ·                  Δ                                ⁢                                                                   ⁢                                  Z                  k                                                                                        (3a)             r:=(x−Xl)·i+(y−Yj)·j+(z−Zk)·k  (3b)
Thus, the integral equation of (2a) is approximated by discrete summation equation (3a). In this form, the summations are made over the complete volume, i.e. the summation is carried out over all volume elements. ΔXi, ΔYj, ΔZk are the dimensions of the individual volume elements. In general, the volume elements will differ in size and accordingly the subscripts i, j and k are used in equations (3a) and (3b) to label the volume elements. The mass of each volume element is equivalent to its volume multiplied by its average density: ρ(i,j,k) ΔXiΔYjΔZk. The size of the volume elements used to model a given total volume, Will affect the accuracy in the determination of the acceleration, with a large number of small elements providing better accuracy than fewer large elements. This forward calculation of the acceleration field from the density can be done easily for any selected point (x,y,z), and is straightforward. Although it typically requires a large amount of computation, this is well within the capability of existing computer technology.
The inverse calculation required to determine the density distribution from the acceleration field is a much more complex computation. There are a number of problems inherent in the approaches that have been attempted to date. The first is that the inversion can be done uniquely only if it is done completely. In other words, a unique inversion is possible only if the density structure is determined for the whole volume that contributes to the observable variations in the gravitational field parameters, i.e., to the variations in the localized gravitational field. The second problem is related to resolution. While it is theoretically possible to perform an inversion to any desired resolution, in practice the resolution is limited by the accuracy and resolution of the measured accelerations. This is a factor in the issue of uniqueness, as the limit in the resolution can be used to limit the volume that needs to be included in the inversion, and allows for defining the solution resolution in such a way that a unique solution is possible.
Any inversion algorithm will result in a set of simultaneous equations that need to be satisfied. It is fundamental that a set of equations can be solved uniquely only if the number of unknowns that need to be determined are equal in number to or fewer than the number of independent observations related to the unknowns. There is a further restriction if there is significant noise in the measurements of a(x,y,z). Noise in the measurements will reduce the resolving power of any instrument, requiring a large number of additional measurements and use of statistical techniques in the inversion. Due to these factors, potential field inversion algorithms in use to date cannot provide a unique inversion because they typically do not properly match the definition of the volume elements to the instrument resolution and do not match the number of independent observations to the number of volume elements.
To date, practitioners have attempted to overcome these limitations by incorporating additional information, typically from other types of measurements (e.g. seismic surveys, magnetometer surveys, etc.). While other types of data can provide additional constraints to assist in the inversion, fundamentally data of another type, i.e. not directly related to density, are not applicable. It is only the density structure that comes into play in setting the gravitational field. No other measurable properties or parameters come into play, as is clear from equations (1) and (2). For example, there is no direct connection between magnetic fields and gravitational fields, and only an indirect connection between seismic data and gravitational data that depends on the relation between seismic wave velocities and density, which is not a unique relationship. All of the above limitations have limited the success of current approaches to the inversion of gravitational data.